Finite groups with \(\Bbb P\)-subnormal biprimary subgroups
V. Kniahina
We consider finite groups only.
Let \(\Bbb P\) be the set of all prime
numbers. A proper subgroup \(H\) of a group \(G\) is called \(\Bbb P\)-subnormal in \(G\)
if there is a chain of subgroups
\[H=H_0\subset H_1\subset \ldots \subset H_n=G\]
such that \(|H_i:H_{i-1}|\) is a prime number for all \(i=1,2,\ldots,n\), see [1].
In [1–3], the finite groups are described with \(\Bbb P\)-subnormal primary subgroups,
2-maximal subgroups, and primary cyclic subgroups, respectively.
In particular, these groups are dispersive, and hence solvable.
We develop this subject by considering the groups with \(\Bbb P\)-subnormal biprimary
dispersive subgroups. The following theorems are proved.
Theorem 1.
Let \(p\) be the greatest prime divisor of the order of \(G\).
If every biprimary \(p\)-closed \(pd\)-subgroup of \(G\) is
\(\Bbb P\)-subnormal in \(G\), then the quotient \(G/O_p(G)\) is \(p\)-nilpotent; in
particular, \(G\) is \(p\)-solvable, and \(l_p(G)\le 2\).
Theorem 2.
Let \(q\) be the smallest prime divisor of the order of \(G\).
If every biprimary \(q\)-nilpotent \(qd\)-subgroup of \(G\) is
\(\Bbb P\)-subnormal in \(G\), then \(G\) is solvable, and \(l_q(G)\le 1\).
Theorem 1 is proved without using the classification of finite simple
groups and, in Theorem 2, the classification is used in the proof of solvability
of a group.
Example 1. We cannot drop the requirement "\(p\) is the greatest prime divisor
of |G|" in the statement of Theorem 1. For instance, consider the
simple group \(PSL(2,11)\). All \(2\)-closed biprimary subgroups of
even order of this group are isomorphic to the alternating
group \(A_4\) of degree 4, which is \(\Bbb P\)-subnormal in \(PSL(2,11)\).
Example 2. The estimate on the \(p\)-length of \(G\) in Theorem 1 is precise.
For instance, all 3-closed biprimary \(3d\)-subgroups of the group \([E_{3^2}]A_4\) are
\(\Bbb P\)-subnormal, and 3-length of this group is equal to 2.
Example 3. A group with \(\Bbb P\)-subnormal \(q\)-nilpotent
biprimary \(qd\)-subgroups may be a simple group for every \(q\ge 3\).
The example for \(q = 3\) is the group \(SL(2,2^n)\) for every odd \(n\ge 3\)
and, for \(q \ge 5\), is the group \(PSL(2,q)\).
Hence we cannot drop the requirement "\(q\) is the smallest prime divisor of \(|G|\)"
in the statement of Theorem 2.
References
A. F. Vasilyev, T. I. Vasilyeva, V. N. Tyutyanov,
On the finite groups of supersoluble type, Sib. Math. J., 51, N6 (2010), 1004–1012.
V. N. Kniahina, V. S. Monakhov, Finite groups with \(\mathbb P\)-subnormal 2-maximal subgroups, ArXiv.org e-Print
archive, arXiv:1105.3663, 18 May 2011.
V. N. Kniahina, V. S. Monakhov, Finite groups with
\(\mathbb P\)-subnormal primary cyclic subgroups, ArXiv.org e-Print archive, arXiv:1110.4720V2, 18 Nov 2011.
